<< Chapter < Page Chapter >> Page >

Solve 3 2 x + 1 = 4 3 x + 1 . State the excluded values.

x = 7 17 . Excluded values are x = 1 2 and x = 1 3 .

Got questions? Get instant answers now!

Solving a rational equation with factored denominators and stating excluded values

Solve the rational equation after factoring the denominators: 2 x + 1 1 x 1 = 2 x x 2 1 . State the excluded values.

We must factor the denominator x 2 −1. We recognize this as the difference of squares, and factor it as ( x 1 ) ( x + 1 ) . Thus, the LCD that contains each denominator is ( x 1 ) ( x + 1 ) . Multiply the whole equation by the LCD, cancel out the denominators, and solve the remaining equation.

( x 1 ) ( x + 1 ) [ 2 x + 1 1 x 1 ] = [ 2 x ( x 1 ) ( x + 1 ) ] ( x 1 ) ( x + 1 ) 2 ( x 1 ) 1 ( x + 1 ) = 2 x 2 x 2 x 1 = 2 x Distribute the negative sign . −3 x = 0 −3 = x

The solution is −3. The excluded values are 1 and −1.

Got questions? Get instant answers now!
Got questions? Get instant answers now!

Solve the rational equation: 2 x 2 + 1 x + 1 = 1 x 2 x 2 .

x = 1 3

Got questions? Get instant answers now!

Finding a linear equation

Perhaps the most familiar form of a linear equation is the slope-intercept form, written as y = m x + b , where m = slope and b = y −intercept . Let us begin with the slope.

The slope of a line

The slope    of a line refers to the ratio of the vertical change in y over the horizontal change in x between any two points on a line. It indicates the direction in which a line slants as well as its steepness. Slope is sometimes described as rise over run.

m = y 2 y 1 x 2 x 1

If the slope is positive, the line slants to the right. If the slope is negative, the line slants to the left. As the slope increases, the line becomes steeper. Some examples are shown in [link] . The lines indicate the following slopes: m = −3 , m = 2 , and m = 1 3 .

Coordinate plane with the x and y axes ranging from negative 10 to 10.  Three linear functions are plotted: y = negative 3 times x minus 2; y = 2 times x plus 1; and y = x over 3 plus 2.

The slope of a line

The slope of a line, m , represents the change in y over the change in x. Given two points, ( x 1 , y 1 ) and ( x 2 , y 2 ) , the following formula determines the slope of a line containing these points:

m = y 2 y 1 x 2 x 1

Finding the slope of a line given two points

Find the slope of a line that passes through the points ( 2 , −1 ) and ( −5 , 3 ) .

We substitute the y- values and the x- values into the formula.

m = 3 ( −1 ) −5 2 = 4 −7 = 4 7

The slope is 4 7 .

Got questions? Get instant answers now!
Got questions? Get instant answers now!

Find the slope of the line that passes through the points ( −2 , 6 ) and ( 1 , 4 ) .

m = 2 3

Got questions? Get instant answers now!

Identifying the slope and y- Intercept of a line given an equation

Identify the slope and y- intercept, given the equation y = 3 4 x 4.

As the line is in y = m x + b form, the given line has a slope of m = 3 4 . The y- intercept is b = −4.

Got questions? Get instant answers now!
Got questions? Get instant answers now!

The point-slope formula

Given the slope and one point on a line, we can find the equation of the line using the point-slope formula.

y y 1 = m ( x x 1 )

This is an important formula, as it will be used in other areas of college algebra and often in calculus to find the equation of a tangent line. We need only one point and the slope of the line to use the formula. After substituting the slope and the coordinates of one point into the formula, we simplify it and write it in slope-intercept form.

The point-slope formula

Given one point and the slope, the point-slope formula will lead to the equation of a line:

y y 1 = m ( x x 1 )

Finding the equation of a line given the slope and one point

Write the equation of the line with slope m = −3 and passing through the point ( 4 , 8 ) . Write the final equation in slope-intercept form.

Using the point-slope formula, substitute −3 for m and the point ( 4 , 8 ) for ( x 1 , y 1 ) .

y y 1 = m ( x x 1 ) y 8 = −3 ( x 4 ) y 8 = −3 x + 12 y = −3 x + 20
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Questions & Answers

the third and the seventh terms of a G.P are 81 and 16, find the first and fifth terms.
Suleiman Reply
if a=3, b =4 and c=5 find the six trigonometric value sin
Martin Reply
pls how do I factorize x⁴+x³-7x²-x+6=0
Gift Reply
in a function the input value is called
Rimsha Reply
how do I test for values on the number line
Modesta Reply
if a=4 b=4 then a+b=
Rimsha Reply
a+b+2ab
Kin
commulative principle
DIOSDADO
a+b= 4+4=8
Mimi
If a=4 and b=4 then we add the value of a and b i.e a+b=4+4=8.
Tariq
what are examples of natural number
sani Reply
an equation for the line that goes through the point (-1,12) and has a slope of 2,3
Katheryn Reply
3y=-9x+25
Ishaq
show that the set of natural numberdoes not from agroup with addition or multiplication butit forms aseni group with respect toaaddition as well as multiplication
Komal Reply
x^20+x^15+x^10+x^5/x^2+1
Urmila Reply
evaluate each algebraic expression. 2x+×_2 if ×=5
Sarch Reply
if the ratio of the root of ax+bx+c =0, show that (m+1)^2 ac =b^2m
Awe Reply
By the definition, is such that 0!=1.why?
Unikpel Reply
(1+cosA+IsinA)(1+cosB+isinB)/(cos@+isin@)(cos$+isin$)
Ajay Reply
hatdog
Mark
jaks
Ryan
how we can draw three triangles of distinctly different shapes. All the angles will be cutt off each triangle and placed side by side with vertices touching
Shahid Reply
Practice Key Terms 7

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Algebra and trigonometry' conversation and receive update notifications?

Ask